Prove by induction that the function $a(n)=5a(n-1)-6a(n-2)$ is equal to $2^{n}+3^{n}$ when a(1)=5 and a(2)=13

Question

I'm having a bit of trouble proving this problem, as I'm not sure what to do at a specific step. For the sake of brevity I'm going to skip the base step where you prove that the original case is true, and move straight to the induction step.

(Assuming that $a(k)=5a(k-1)-6a(k-2)=2^{k}+3^{k}$)

1. $5a((k+1)-1)-6a((k+1)-2)$
2. $5a(k)-6a(k-1)$
3. $5[5a(k-1)-6a(k-2)]-6a(k-1)$
4. $25a(k-1)-30a(k-2)-6a(k-1)$

I'm not entirely sure where to go after this step (or even if that last step is sensible), I don't actually want an answer, but would prefer if you could guide me along (or at least show me if I went wrong somewhere).

Thanks in advance

Answer 1

hint: $a_{n+1} = 5a_n-6a_{n-1}=5(2^n+3^n)-6(2^{n-1}+3^{n-1})=5(2^n+3^n)-3\cdot 2^n-2\cdot 3^n=....$